# Are there irreducible multi-fusion categories that are not fusion categories?

Multi-fusion categories are a generalization of fusion categories with a non-simple unit. The direct sum of two multi-fusion categories is again a multi-fusion category. By irreducible I mean that a multi-fusion category cannot be written as a non-trivial direct sum. Are all such irreducible multi-fusion categories fusion categories?

Bonus question: Are there multi-fusion categories that are not Morita equivalent to a direct sum of fusion categories?

[I'm mainly interested in the unitary case.]

Matrix categories, $$\mathrm{End}(\mathrm{Vec}^{\oplus n})$$. (The identity on each copy of $$\mathrm{Vec}$$ are summands of the identity.)
• Assuming the base field is C (weird issues arise if it's not algebraically closed), I'm pretty sure any multifusion category is Morita equivalent to a fusion category. Take $1_i$ to be a summand of $1$, and look at the category of $1_i$-modules. This gives a Morita equivalence between your original category and the fusion category of $1_i$-bimodules. This is a rephrasing of my parenthetical. – Noah Snyder Mar 6 '19 at 19:04